Question

Mala travels 300 km to her home partly by train and partly by bus. She takes 4 hours if she travels 60km by train and the remaining by bus. If she travels 100 km by train and the remaining by bus, she takes 10 minutes longer. Find the speed of the train and the bus separately.

Answer 1

Hint: This is a two variable linear equation question. From the given data first we will derive two linear equations containing two variables. Solving both will get the speed of the train and bus.

Complete step-by-step answer:
The total distance Mala has to travel = 300km
She has to travel partly by train and partly by bus.
Let the speed of the train be x km per hour and speed of the bus be y km per hour respectively.
According to the question,
Mala takes 4 hours if she travels 60km by train and the remaining by bus.
For this case, distance travelled by train = 60km
Hence the distance travelled by the bus $ = 300 - 60 = 240 $ km
Putting the above data into a linear equation as per the formula, ‘time = distance / speed’ we get,
 $ \dfrac{{60}}{x} + \dfrac{{240}}{y} = 4 $ ………………… (1)
Again it is given that if she travels 100km by train and the remaining by bus, she takes more 10 minutes i.e. $ \dfrac{{10}}{{60}} = \dfrac{1}{6} $ hours more.
For this case, distance travelled by train = 100km
Hence the distance travelled by the bus $ = 300 - 100 = 200 $ km
Putting the above data into a linear equation as per the formula, ‘time = distance / speed’ we get,
 $ \dfrac{{100}}{x} + \dfrac{{200}}{y} = 4 + \dfrac{1}{6} $
 $ \dfrac{{100}}{x} + \dfrac{{200}}{y} = \dfrac{{25}}{6} $ ……………….. (2)
Let’s take $ \dfrac{1}{x} = u $ and $ \dfrac{1}{y} = v $
Substituting this in the equation 1 and 2 we get,
 $ 60u + 240v = 4 $ ……………………… (3)
And, $ 100u + 200v = \dfrac{{25}}{6} $
By cross multiplication we get,
 $ 600u + 1200v = 25 $ ………………………….(4)
Multiplying equation 3 by 10 we get,
 $ 600u + 2400v = 40 $ …………….. (5)
Subtracting equation 4 from equation 5 we get,
 $ (600u + 2400v) - (600u + 1200v) = 40 - 25 $
Expanding the above equation and simplifying it we get,
 $ 1200v = 15 $
Taking 1200 to the right hand side we get,
 $ v = \dfrac{{15}}{{1200}} = \dfrac{1}{{80}} $
Putting the value of v in equation 3 we get,
 $ 60u + 240 \times \dfrac{1}{{80}} = 4 $
Simplifying the above equation we get,
 $ 60u + 3 = 4 $
Hence, $ u = \dfrac{1}{{60}} $
Earlier we have taken that $ \dfrac{1}{x} = u $ and $ \dfrac{1}{y} = v $
Hence, $ u = \dfrac{1}{x} = \dfrac{1}{{60}} $
By cross multiplication we get,
 $ x = 60 $ km per hour
And $ v = \dfrac{1}{y} = \dfrac{1}{{80}} $
By cross multiplication we get,
 $ y = 80 $ km per hour
 $ \therefore $ Thus the speed of the train and the speed of the bus are 60 km per hour and 80 km per hour respectively.

Note: A linear equation in two variables can be written as $ ax + by = c $ where x and y are two variables.
Two equations with the same variable are called systems of equations.
There are 3 methods to solve the system of linear equations such as substitution method, elimination method and addition method. Other than these we can also use graphing methods.
In such a type of question, the main thing is to form linear equations from given data assuming some variables.